Super-harmonic resonance of fractional-order van der Pol oscillator

Wei Peng; Shen Yong-Jun; Yang Shao-Pu

doi:10.7498/aps.63.010503

Abstract

The dynamical characteristics of super-harmonic resonance of van der Pol oscillator with fractional-order derivative are studied. First the approximate analytical solution are obtained by the averaging method, and the definitions of equivalent linear damping and equivalent linear stiffness for super-harmonic resonance are established. Effects of the fractional-order parameters on the dynamical characteristics of the system are also studied through the equivalent linear damping and equivalent linear stiffness. Moreover, the amplitude-frequency equation and the stability condition for the steady-state solution are analytically presented, and the definitions of equivalent nonlinear damping coefficient and nonlinear stability parameter are also established. Finally, the comparisons of the fractional-order and the traditional integer-order van der Pol oscillators are carried out by numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also analyzed.

Keywords:

Authors and contacts

1.
Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11072158, 11372198), and the Program for New Century Excellent Talents in University of Ministry of Education of China (Grant No. NCET-11-0936).

References

[1]	Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[2]	Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 150503]
[3]	Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[4]	Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Non-Linear Mech. 47 975
[5]	Gorenflo R, Abdel R E A 2007 J. Comput. Appl. Mathe. 205 871
[6]	Jumarie G 2006 Compu. Mathe. Appl. 51 1367
[7]	Ishteva M, Scherer R, Boyadjiev L 2005 Mathe. Sciences Research J. 9 161
[8]	Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523
[9]	Wang Z H, Du M L 2011 Shock Vib. 18 257
[10]	Wang Z H, Hu H Y 2009 Science in China Series G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学力学天文学 39 1495]
[11]	Wang Z H, Hu H Y 2010 Sci. Chin. Phys. Mech. Astron. 53 345
[12]	Shi M, Wang Z H 2013 Science Sinica: Phys. Mech. Astron. 43 467 (in Chinese) [石敏, 王在华 2013 中国科学: 物理学力学天文学 43 467]
[13]	Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 62 024501]
[14]	Gu R C, Xu Y, Zhang H Q, Sun Z K 2011 Acta Phys. Sin. 60 110514 (in Chinese) [顾仁财, 许勇, 张慧清, 孙中奎 2011 60 110514]
[15]	Cao J X, Ding H F, Li C P 2013 Commun. Appl. Mathe. Comput. 27 61 (in Chinese) [曹建雄, 丁恒飞, 李常品 2013 应用数学与计算数学学报 27 61]
[16]	Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2011 62 140503]
[17]	Chen L C, Zhu W Q 2009 J. Vib. Control. 15 1247
[18]	Huang Z L, Jin X L 2009 J. Sound Vib. 319 1121
[19]	Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]
[20]	Zhou C Y, Li G L, Zhang C, Chi B Y, Li D M, Wang Z H 2009 J. Semicon. 30 075008
[21]	Wu R C, Hei X D, Chen L P 2013 Commun. Theor. Phys. 60 189
[22]	Zhou G Q, Wang X G, Chu X X 2013 Science China: Phys. Mech. Astron. 56 1487
[23]	Zeng F H, Li C P 2013 Chin. J. Comput. Phys. 30 491 (in Chinese) [曾凡海, 李常品 2013 计算物理 30 491]
[24]	Li C P, Zhao Z G 2009 J. Shanghai Univ. (Engl. Ed.) 13 197 (in Chinese) [李常品, 赵振刚 2009 上海大学学报(英文版) 13 197]
[25]	Hu J B, Zhao L D, Xie Z G 2013 Chin. Phys. B 22 080506
[26]	Rajneesh K, Vandana G 2013 Chin. Phys. B 22 074601
[27]	Tian Y S 2013 Acta. Mathe. Appl. Sin. 29 661
[28]	Liu D, Xu W, Xu Y 2013 Acta. Mech. Sin. 29 443
[29]	Lan Y H, Li W J, Zhou Y, Luo Y P 2013 Inter. J. Auto. Comput. 10 296
[30]	Kumar R, Gupta V 2013 Chin. Phys. B 22 074601
[31]	Wang H Q 1992 Nonlinear Vibration (Bei Jing: Higher Education Press) p131 (in Chinese) [王海期 1992 非线性振动 (北京: 高等教育出版社) 第131页]
[32]	Leung A Y T, Yang H X, Guo Z J 2012 J. Sound Vib. 331 1115
[33]	Sardar T, Ray S S, Bera R K, Biswas B B 2009 Phys. Scr. 80 025003
[34]	Xie F, Lin X Y 2009 Phys. Scr. 136 014033
[35]	Chu Y Q, Li C Y 1996 Analysis of Nonlinear Vibrations (Beijing: Beijing Institute of Technology Press) pp828–832 (in Chinese) [褚亦清, 李翠英 1996 非线性振动分析 (北京: 北京理工大学出版社) 第828–832页]

Cited By

[1]	Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[2]	Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 150503]
[3]	Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[4]	Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Non-Linear Mech. 47 975
[5]	Gorenflo R, Abdel R E A 2007 J. Comput. Appl. Mathe. 205 871
[6]	Jumarie G 2006 Compu. Mathe. Appl. 51 1367
[7]	Ishteva M, Scherer R, Boyadjiev L 2005 Mathe. Sciences Research J. 9 161
[8]	Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523
[9]	Wang Z H, Du M L 2011 Shock Vib. 18 257
[10]	Wang Z H, Hu H Y 2009 Science in China Series G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学力学天文学 39 1495]
[11]	Wang Z H, Hu H Y 2010 Sci. Chin. Phys. Mech. Astron. 53 345
[12]	Shi M, Wang Z H 2013 Science Sinica: Phys. Mech. Astron. 43 467 (in Chinese) [石敏, 王在华 2013 中国科学: 物理学力学天文学 43 467]
[13]	Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 62 024501]
[14]	Gu R C, Xu Y, Zhang H Q, Sun Z K 2011 Acta Phys. Sin. 60 110514 (in Chinese) [顾仁财, 许勇, 张慧清, 孙中奎 2011 60 110514]
[15]	Cao J X, Ding H F, Li C P 2013 Commun. Appl. Mathe. Comput. 27 61 (in Chinese) [曹建雄, 丁恒飞, 李常品 2013 应用数学与计算数学学报 27 61]
[16]	Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2011 62 140503]
[17]	Chen L C, Zhu W Q 2009 J. Vib. Control. 15 1247
[18]	Huang Z L, Jin X L 2009 J. Sound Vib. 319 1121
[19]	Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]
[20]	Zhou C Y, Li G L, Zhang C, Chi B Y, Li D M, Wang Z H 2009 J. Semicon. 30 075008
[21]	Wu R C, Hei X D, Chen L P 2013 Commun. Theor. Phys. 60 189
[22]	Zhou G Q, Wang X G, Chu X X 2013 Science China: Phys. Mech. Astron. 56 1487
[23]	Zeng F H, Li C P 2013 Chin. J. Comput. Phys. 30 491 (in Chinese) [曾凡海, 李常品 2013 计算物理 30 491]
[24]	Li C P, Zhao Z G 2009 J. Shanghai Univ. (Engl. Ed.) 13 197 (in Chinese) [李常品, 赵振刚 2009 上海大学学报(英文版) 13 197]
[25]	Hu J B, Zhao L D, Xie Z G 2013 Chin. Phys. B 22 080506
[26]	Rajneesh K, Vandana G 2013 Chin. Phys. B 22 074601
[27]	Tian Y S 2013 Acta. Mathe. Appl. Sin. 29 661
[28]	Liu D, Xu W, Xu Y 2013 Acta. Mech. Sin. 29 443
[29]	Lan Y H, Li W J, Zhou Y, Luo Y P 2013 Inter. J. Auto. Comput. 10 296
[30]	Kumar R, Gupta V 2013 Chin. Phys. B 22 074601
[31]	Wang H Q 1992 Nonlinear Vibration (Bei Jing: Higher Education Press) p131 (in Chinese) [王海期 1992 非线性振动 (北京: 高等教育出版社) 第131页]
[32]	Leung A Y T, Yang H X, Guo Z J 2012 J. Sound Vib. 331 1115
[33]	Sardar T, Ray S S, Bera R K, Biswas B B 2009 Phys. Scr. 80 025003
[34]	Xie F, Lin X Y 2009 Phys. Scr. 136 014033
[35]	Chu Y Q, Li C Y 1996 Analysis of Nonlinear Vibrations (Beijing: Beijing Institute of Technology Press) pp828–832 (in Chinese) [褚亦清, 李翠英 1996 非线性振动分析 (北京: 北京理工大学出版社) 第828–832页]

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Vol. 63, Issue 1 - 2014-01-05

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